Fraïssé structures with universal automorphism groups

Journal of Algebra 463 (2016) 134–151

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Journal of Algebra www.elsevier.com/locate/jalgebra

Fraïssé structures with universal automorphism groups Isabel Müller a r t i c l e

i n f o

Article history: Received 20 January 2015 Available online 1 July 2016 Communicated by Martin Liebeck Keywords: Universal groups Automorphism groups Polish groups Fraïssé structures Tower constructions Urysohn space Generalized polygons Independence relations Combinatorics

a b s t r a c t We prove that the automorphism group of a Fraïssé structure M equipped with a notion of stationary independence is universal for the class of automorphism groups of substructures of M . Furthermore, we show that this applies to certain homogeneous n-gons. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Certain homogeneous structures are universal with respect to the class of their substructures: The Rado graph is universal for the class of all countable graphs, the rationals as a dense linear order for the class of all countable linear orders and Urysohn’s universal Polish space for the class of all Polish spaces. Jaligot asked whether a universal structure M transfers its universality onto its automorphism group, i.e. whether Aut(M ) is universal for the class of automorphism groups of substructures of M (cf. [6]). Recently, Doucha showed that, for an uncountable structure M , the answer to Jaligot’s question is E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2016.06.010 0021-8693/© 2016 Elsevier Inc. All rights reserved.

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rarely positive [3]. Countable homogeneous structures on the contrary, most often have universal automorphism groups. In fact, the only known counterexample was pointed out by Piotr Kowalski and is given by the Fraïssé limit of finite fields in fixed characteristic p, ˆ which is which coincides with the algebraic closure of Fp . Its automorphism group is Z, torsion free and hence does not embed any automorphism group of a finite field. It is still unknown if there is a relational countable counterexample. We will prove that in the case where M is a Fraïssé structure admitting a certain stationary independence relation, the automorphism group Aut(M ) will be universal for the class of automorphism groups of substructures of M . Uspenskij [11], using a careful construction of Urysohn’s universal Polish space given by Katětov [7], proved that its isometry group is universal for the class of all Polish groups, which corresponds to the class of isometry groups of Polish spaces [4]. The idea of Katětov thereby can be described as follows: Given a Polish space X, he constructed a new metric space E1 (X) consisting of X together with all possible 1-point metric extensions, while assigning the smallest possible distance between new points. Under minor restrictions, the space obtained is again Polish, denoted by the first Katětov space of X. Iterating this, i.e. building one Katětov space over the other, he constructed a copy of Urysohn’s space itself. Furthermore, all isometries of X extend in a unique way at every step of the construction, which yields the desired embedding of Isom(X) into Isom(U). In [2] Bilge adapted this construction to Fraïssé limits of rational structures with free amalgamation by gluing extensions freely over the given space. Both Urysohn’s spaces and Fraïssé classes with free amalgamation carry an independence relation as introduced by Tent and Ziegler [10]. In this paper, we will show that the mere presence of a stationary independence relation within a Fraïssé structure M allows us to mimic Katětov’s construction of Urysohn’s universal metric space, starting with any structure X embeddable in M . With the help of the given independence relation, we will glue “small” extensions of X independently and construct an analog of Katětov spaces in the non-metric setting, thereby ensuring that the automorphisms of X extend canonically to its Katětov spaces and that the extensions behave well under composition. In particular, we will give a positive answer to the question of Jaligot for the class of Fraïssé limits with stationary independence relation by proving the following result (Theorem 4.9): Theorem. Let M be a Fraïssé structure with stationary independence relation and Kω the class of all countable structures embeddable into M . Then for any X ∈ Kω , there is an embedding f : X → M such that every automorphism of f (X) extends to an automorphism of M in such a way that this extension yields a continuous embedding of Aut(X) into Aut(M ). In particular, the automorphism group Aut(M ) is universal for the class Aut(Kω ) := {Aut(X) | X ∈ Kω }, i.e. every group in Aut(Kω ) can be continuously embedded as a subgroup into Aut(M ). Note, that every automorphism group of a countable first order structure M can be considered as a Polish group if we equip it with the topology of pointwise convergence.

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The basic open sets for that topology Ou := {f ∈ Aut(M ) | f |A = u} are determined by finite partial isomorphisms u : A → M , where A ⊆ M is a finite subset of M . 2. Preliminaries Let us briefly recall the central concepts of Fraïssé theory used in the article. For further reading and proofs in this topic, there is a plethora of sources, see for example [5, p. 158ff.] or [9, p. 69ff.]. Let L be a countable language and K a class of finitely generated L-structures which is countable up to isomorphism types. We call K a Fraïssé class if the following three conditions are satisfied: HP For any finitely generated L-structure A which is embeddable into some B ∈ K, there is a structure A in K isomorphic to A. JEP For every B and C in K, there is some D ∈ K such that both B and C are embeddable into D. AP For every A, B and C in K together with embeddings f1 : A → B and f2 : A → C, there is some D in K together with embeddings g1 : B → D and g2 : C → D such that the following diagram commutes:

f1

B

g1

A

D. f2

C

g2

We call the class of all finitely generated substructures of an L-structure M the skeleton of M . An L-structure M is called rich with respect to a class K of finitely generated L-structures if, for all A and B in K together with embeddings f : A → B and g : A → M , there is an embedding h : B → M such that h ◦ f = g. Finally, we say that an L-structure is K-saturated if its skeleton is exactly K and it is furthermore rich with respect to K. The following fact is the main theorem of Fraïssé theory: Fact 2.1. Assume K to be a class of finitely generated L-structures, countable up to isomorphism types. Then there is a countable K-saturated structure M if and only if K is a Fraïssé class. Furthermore, any two countable K-saturated structures are isomorphic.

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Given a Fraïssé-class K, the corresponding K-saturated structure as above is called the Fraïssé limit of K. Note that a countable structure M is the Fraïssé limit of its skeleton if and only if M is homogeneous, i.e. every partial isomorphism between finitely generated substructures can be extended to an automorphism of M . We call such structures Fraïssé structures. 3. Stationary independence The main ingredient to generalize Katětov’s construction to arbitrary Fraïssé structures is the presence of a stationary independence relation. For the following a, b, . . . denote finite tuples, by A, B, C, . . . we denote small, i.e. finitely generated structures, whereas X, Y, . . . stand for arbitrary countable ones. Given substructures A and B, the substructure generated by their union is denoted by AB. By a type over X, we mean a set of L(X) formulas p(x) with free variables x which is maximal satisfiable in M . The type tp(a/X) of a tuple a over X is the set of all L(X) formulas which are satisfied by a. Definition 3.1 ((Local) stationary independence relation). Assume M to be a homoge| on the finitely generated substructures of M is neous L-structure. A ternary relation  called a stationary independence relation (SIR) if the following conditions are satisfied: SIR1 (Invariance). The independence of finitely generated substructures in M only de| CB pends on their type. In particular, for any automorphism f of M , we have A  | f (C) f (B). if and only if f (A)  | C B, then B  | C A. SIR2 (Symmetry). If A  | C BD, then A  | C B and A  | BC D. SIR3 (Monotonicity). If A   SIR4 (Existence). For any A, B and C in M , there is some A |= tp(A/C) with A  | C B.  SIR5 (Stationarity). If A and A have the same type over C and are both independent over C from some set B, then they also have the same type over BC. If the relation A  | C B is only defined for nonempty C, we call  | a local stationary independence relation.  Remark 3.2. Any SIR also fulfills the following property, which was part of the original definition in [10]: SIR6 (Transitivity). If A  | C B and A  | BC D, then A  | C BD. To see that, consider A, B, C and D with A  | C B and A  | BC D. We have to show that this implies the independence of A and BD over C. By Existence there is some A ≡C A with A  | C BD. By Monotonicity and Stationarity we get A ≡BC A. Again by Monotonicity and Stationarity, we obtain A ≡BCD A and hence A  | C BD, as desired. 

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For a (local) SIR  | defined on some homogeneous structure M , we call the pair (M,  | ) a (local) SI-structure. If the interpretation of  | in M is clear or irrelevant, we will refer to M alone as an SI-structure. Remark 3.3. If A and A in M have the same quantifier free (qf-)type over some B ⊆ M , then the map AB → A B is a partial isomorphism. In homogeneous structures such a map extends to an automorphism of the whole structure M , fixing B and sending A to A . As we will exclusively work inside homogeneous structures for the rest of the article, note that A and A have the same qf-type over B (denoted by tpqf (A/B) = tpqf (A /B)) if and only if there is an automorphism of M that fixes B pointwise and maps A to A (write A ≡B A ). A necessary condition for a given structure to carry a stationary independence relation is given by the following fact (cf. [10], Proof of Lemma 5.1). Fact 3.4. Algebraic and definable closures coincide in an SI-structure M , i.e. acl(X) = dcl(X) for all X ⊂ M . Note that Fact 3.4 also holds in local SI-structures for nonempty X. Furthermore, the equality acl(∅) = dcl(∅) is true if and only if either acl(∅) = ∅ or every automorphism has a fixed point. This characterization of algebraic closures in SI-structures already implies that we cannot define a stationary independence relation on every Fraïssé structure: The class of finite fields in a fixed characteristic p forms a Fraïssé class. In its limit, the algebraically ¯ p of characteristic p, algebraic and definable closure differ. Thus Fact 3.4 closed field F ¯ p . As mentioned states that no stationary independence relation can be defined on F ˆ which does not embed any before, its automorphism group is the torsionfree group Z, ¯ finite group. Thus, the group Aut(Fp ) is not universal for the class of automorphism ¯ p. groups of substructures of F On the other hand, the rationals as a dense linear order form another example of a Fraïssé structure which does not allow a notion of stationary independence, but still has a universal automorphism group. These two examples show that the absence of a notion of stationary independence within a Fraïssé structure does not decide about the universality of its automorphism group. Nevertheless, several examples of Fraïssé structures admitting a stationary independence relation are known. Amongst them are the rational Urysohn space and the Urysohn sphere as well as Fraïssé limits of rational free amalgamation classes [2]. More examples, also including a non-relational SI-structure, will be discussed in detail in section 5. Unlike forking in simple theories, which is uniquely determined by its properties, a Fraïssé structure can carry different notions of stationary independence: For an example, consider the random graph, and define finite subgraphs A and B to be independent

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over some finite subgraph C if and only if A ∩ B ⊆ C and every vertex in A \ C is connected to every vertex in B \ C. It is not hard to verify that this defines a stationary independence relation. On the other hand, the class of finite graphs is a free amalgamation class, whence another stationary independence relation is given by the free amalgam of A and B over C, i.e. A and B are defined to be independent over C if and only if A ∩ B ⊆ C and no vertex in A \ C is connected to a vertex in B \ C. We have to develop some tools to mimic Katětov’s construction of Urysohn’s space. In order to merge certain small extensions over any embeddable infinite substructure in an independent way, we will need to extend the independence notion to arbitrary base sets: Definition 3.5. Let M be an SI-structure. Two substructures A and B are independent over X ⊆ M (write A  | X B), if and only if there is some finitely generated C ⊂ X such that A  | C  B for every finitely generated C  ⊂ X containing C. Notice that in the examples of SI-structures mentioned above, the independence relation is naturally defined between arbitrary sets and coincides with the one given in the previous definition. Lemma 3.6. The independence relation  | extended to arbitrary base sets satisfies all the properties of an SIR except possibly Existence. Proof. Invariance and Monotonicity easily follow from the definition. To see Transitivity, assume X ⊆ M and finitely generated A, B and D ⊂ M given with A | B and A  | D. XB

X

By definition, there are C1 and C2 ⊂ X such that C1 and C2 B are finite supports for the first and the second independence respectively, i.e. A  | C  B (resp. A  | C  D) for every finitely generated C  ⊂ X (resp. C  ⊂ XB) containing C1 (resp. C2 B). If we set C := C1 C2 , then every finitely generated C  ⊂ X containing C satisfies A | B and A  | D, hence A  | BD,    C

C B

C

which yields Transitivity. To prove Stationarity, note that two different realizations of the same type over X, both independent from some finite set B, have different types over XB if and only if there is some finite subset C  of X such that their types differ already over C  B. 2 The homogeneity of M allows us furthermore to speak of independence between subsets of embeddable structures. We denote by Kω the class of all structures embeddable

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into M , i.e. the class of all structures whose skeleton is contained in the skeleton of M . For some Y ∈ Kω with substructures A, B and X, we say that A is independent from B over X if the same is true for one, and hence for every embedding of Y into M . 4. A general Katětov construction Since the independence relation is defined only in one model and need not be part of the theory, we cannot ensure the existence of independent extensions for types over base sets which are not necessarily finitely generated. Nevertheless, a variant of Existence for certain types, called finitely supported, can be deduced. Definition 4.1. Let (M,  | ) be an SI-structure, and AX arbitrary in Kω . We say that AX is finitely supported (over X), if there is a finitely generated subset C ⊆ X such that A  | C D for all finitely generated D ⊆ X. In that case we also write A  | C X and refer to C as a support of AX over X. Furthermore, we call a quantifier free type π(x) over X finitely supported, if it defines a finitely supported Kω -structure. Loosely speaking, a type π(x) over X is finitely supported if its realizations are independent from the base set over some finitely generated substructure C of X. It is not hard to see that every finitely generated D ⊆ X that contains C is again a support for π(x), so that for any finite family of finitely supported types over X we can choose a common support. Let us denote by S qf (X) the set of all quantifier-free types over X. In the following, we show some useful properties of finitely supported types and structures. Lemma 4.2. Assume X to be a Kω -structure. i) Suppose C ⊆ X is finitely generated and π := π(x) a qf-type over C realized in M . Then π has a unique extension π ˜ ∈ S qf (X) which is finitely supported over X with support C. ii) Let AX and BX ∈ Kω be finitely supported over X. Then there is some A B  X   ∈ Kω with A X   ∼ | B. = AX, B  X   ∼ = BX and A   X

Furthermore, the structure A B  X   is again finitely supported over X  ∼ = X. Proof. By choosing an arbitrary embedding, we may assume X to be a substructure of M .  i) Write X as the limit of a chain X = n∈ω Cn with C0 := C. Inductively we can construct a chain of types π0 ⊆ π1 ⊆ . . . by setting π0 := π and for each n > 0,

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we set πn := tpqf (An /Cn ), where An ⊆ M with An |= πn−1 and An  | C Cn . n−1  By compactness and Transitivity, the set π ˜ := n∈ω πn is a finitely supported type over X with support C. Note that π ˜ defines again a Kω -structure AX, as every finite subset of AX is embeddable in some An Cn  ⊆ M . The uniqueness of π ˜ now follows from Stationarity. ii) Let C ⊂ X be some common support of AX and BX over X. By Existence we find realizations A1 (resp. B1 ) of the qf-type of A (resp. B) over C in M such that A1  | C B1 . Part i) allows us to extend the type tpqf (A1 B1 /C) to some finitely supported type π over X with support C, which defines a Kω -structure A2 B2 X. As A2 (resp. B2 ) and A (resp. B) have the same qf-type over C and are both independent from X over C, Stationarity implies that AX ∼ = A2 X (resp. BX ∼ = B2 X).  Furthermore, for any finitely generated C ⊂ X containing C we have A2 B2   | C  and A2  | B2 . C

C

So A2  | C  B2 by Monotonicity and Transitivity, and thus A2  | X B2 .

2

The second part of the above lemma shows how to independently glue certain structures over arbitrary Kω -base sets. As we will see below, the Kω -structure described in Lemma 4.2.ii) is unique up to isomorphism. This justifies the following definition. Definition 4.3. Assume AX and BX to be finitely supported Kω -structures. The structure A B  X   ∈ Kω obtained in Lemma 4.2.ii) is called the SI-amalgam of AX and BX over X and denoted by A ∗X B. Since an SI-amalgam is again a finitely supported Kω -structure, we can amalgamate finite families of finitely supported Kω -structures. This process behaves well under permutations of the given finite family. Lemma 4.4. The SI-amalgam of two structures is unique up to isomorphism. Moreover, SI-amalgamation is commutative and associative, meaning that for given finitely supported AX, BX and CX, the SI-amalgams A ∗X B and B ∗X A (resp. (A ∗X B) ∗X C and A ∗X (B ∗X C)) are isomorphic. Proof. Let two Kω -structures A1 B1 X1  and A2 B2 X2  be given with Ai Xi ∼ = AX and ∼ Bi Xi = BX as well as Ai  | X Bi . By Lemma 4.2.ii), the structures Ai Bi Xi  are again i finitely supported with support Ci ⊆ Xi . Because Ai Xi ∼ = AX, we may assume that ∼ Ai Ci = AC for some C ⊂ X. Note that we can pick the Ci ’s such that they also witness the independence of Ai and Bi over Xi , i.e. Ai  | C  Bi for all C  ⊂ Xi with  Ci ⊆ C . By Stationarity and Invariance, it suffices to show that A1 B1 C1 ∼ = A2 B2 C2 . Let us assume that the structures are embedded into M . By homogeneity, the partial isomorphism f : A1 C1 → A2 C2 extends to A1 B1 C1 , yielding a copy f (B1 ) := B2 of B1 .

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The structures B2 and B2 have the same type over C2 and are both independent from A2 over it. Hence, there is an automorphism g ∈ Aut(M ) that fixes A2 C2 and sends B2 to B2 . Finally, the map g ◦ f : A1 B1 C1  → A2 B2 C2  provides the desired isomorphism. Commutativity follows directly from Symmetry of our independence relation. It remains to show that the amalgamation process is associative. To see that, assume A1 B1 C1 X1  = A ∗X (B ∗X C). By definition of the SI-amalgam, we have A1  | X B1 C1 1 and B1  | X C1 , whence 1

(1) A1  | B1 and (2) A1 B1  | C1 , X1

X1

by Monotonicity and Transitivity. Now (1) yields A1 B1 X1  = A ∗X B, whereas (2) concludes that A1 B1 C1 X1  = (A ∗X B) ∗X C. This implies A ∗X (B ∗X C) ∼ = (A ∗X B) ∗X C, as desired. 2 Lemma 4.4 guarantees that the order in which we amalgamate a finite family of structures is irrelevant. Hence, for a given finite family of finitely supported Kω -structures {Ai X, i ∈ n}, it makes sense to write: ∗X A i i∈n

:= ((. . . ((A0 ∗X A1 ) ∗X A2 ) . . .) ∗X An−1 ).

This amalgamation will be the main tool for developing a general analog of the socalled Katětov spaces in the non-metric setting. When we now move on to countable families F := {Ai X, i ∈ ω} of finitely supported structures over X, note that every finite amalgam ∗X Ai can naturally be embedded into ∗X Ai , so that the family i≤n

i≤n+1

(∗X Ai )i∈ω is a directed system. The structure ∗X Ai generated by the limit of this sysi∈ω

i≤n

tem is still a Kω -structure, called the SI-amalgam of {Ai X, i ∈ ω}. As we are mainly interested in extensions of automorphisms, the following lemma will be useful further on. Lemma 4.5. Let {Ai X, i ∈ ω} be a countable family of Kω -structures, finitely supported over X. Let furthermore {fi : Ai X → Aσ(i) X, i ∈ ω} be a family of isomorphisms, where  σ is a permutation of ω and fi |X = fj |X for all i, j. Then the union i∈ω fi induces an automorphism of the SI-amalgam ∗X Ai . i∈ω

Proof. We will establish the statement for the SI-amalgam of two structures. The claim follows via induction. Hereby, surjectivity in the limit process is given by the surjectivity of σ. As all of the structures are in Kω , we may take A0 ∗X A1 and Aσ(0) ∗X Aσ(1) to be substructures of M and hence the fi ’s to be partial isomorphisms. It suffices to show

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that g := f0 ∪ f1 restricted to any finitely generated substructure of A0 ∗X A1 is again a partial isomorphism. Choose D ⊂ X arbitrary. As g|A0 D = f0 |A0 D , the restriction of g to A0 D defines a partial isomorphism between finitely generated substructures of M and hence it extends to an automorphism g˜ ∈ Aut(M ). Denote by Bσ(1) the image of A1 under g˜. Then Aσ(1) and Bσ(1) have the same type over D, as g˜|D = f0 |D = f1 |D , and are both independent from Aσ(0) over D by Invariance. Stationarity implies now that they even have the same type over DAσ(0) , whence there is some h ∈ Aut(M ) fixing DAσ(0) and sending Bσ(1) to Aσ(1) . Thus h˜ g|A0 A1 D = g|A0 A1 D is a partial isomorphism and we are done. 2 The above lemma yields a crucial property of SI-amalgams needed to imitate Katětov’s construction. We can now define a chain of Katětov spaces in this setting. Given an arbitrary Kω -structure X, denote by S fin (X) the space of all finitely supported qf-types over X. The set S fin (X) gives rise to a countable family F of finitely supported structures and, after fixing an arbitrary enumeration, we can build the SIamalgam of that family. Definition 4.6. After choosing an arbitrary enumeration S fin(X) = {Ai X | i ∈ ω} of the space of qf-types over some Kω -structure X, let E1 (X) := ∗X Ai be the SI-amalgam i∈ω

of that family and call it the first Katětov space of X. One can show that E1 (X) does not depend on the chosen enumeration of S fin (X). Moreover, the space E1 (X) is again a Kω -structure, whence we may iterate the procedure and thereby construct inductively the n-th Katětov spaces En (X) of X as follows: E0 (X) := X, En+1 (X) := E1 (En (X)). This family of Katětov spaces comes equipped with natural embeddings between its members and therefore it forms an inductive system. The limit E(X) of that system will be called the Katětov limit of X. In order to prove Theorem 4.9, it remains to show that the Katětov limit of an arbitrary Kω (M )-structure X is isomorphic to M and Aut(X) embeds continuously into Aut(E1 (X)). Lemma 4.7. Assume M to be an SI-structure and consider an arbitrary X ∈ Kω (M ). The Katětov limit E(X) is isomorphic to M . Proof. Let K be the skeleton of M . As M is homogeneous, the class K is a Fraïssé class and M = Fr(K) is K-saturated. Any two countable K-saturated structures are isomorphic, whence it suffices to prove K-saturation for E(X) to establish the lemma.

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We will first show that K is exactly the skeleton of E(X). It is easy to see that K is contained in the skeleton, as for any finitely generated structure A = a of M , the type tpqf (a/∅) determines completely A and it can be extended to a finitely supported type over X by Lemma 4.2.i). Hence, there is a copy of each structure from K inside the first Katětov space E1 (X), and thus also in the limit E(X). For the other direction, let A ⊂ E(X) be an arbitrary finitely generated substructure. Then there is some n ∈ ω such that A ⊂ En (X). Since all the Katětov-spaces En (X) are Kω -structures, it follows that A ∈ K by definition, so K = K(E(X)) as desired. It remains to show that E(X) is rich with respect to K. Consider some finitely generated substructure A ⊂ E(X) and a K-structure B with f : A → B = Ab. Again, the structure A is contained in some En (X) and we can extend tpqf (b/A) to a finitely supported type over En (X). Since a realization of this type occurs in En+1 (X), we can embed B in E(X) over A. Consequently, the countable Katětov-limit of an arbitrary Kω -structure X is K-saturated and hence isomorphic to M . 2 Lemma 4.8. Let M be an SI-structure and X ∈ Kω arbitrary. Then every automorphism in Aut(X) can be canonically extended to an automorphism of E1 (X) such that the extension yields a continuous embedding of Aut(X) into Aut(E1 (X)). Moreover, the group Aut(X) embeds continuously into Aut(E(X)). Proof. Consider an automorphism f ∈ Aut(X) and observe that f induces a permutation σf of S fin (X) = (Ai X | i ∈ ω), the space of all finitely supported qf-types over X. By Lemma 4.5, this gives rise to an automorphism of the amalgam ∗i∈ω Ai = E1 (X). Hence, for every f ∈ Aut(X), there is an automorphism fˆ ∈ Aut(E1 (X)) which extends f . As for every finitely supported type π(x) over X there exists a unique k ∈ ω with Ak |= π(x), these extensions behave well under multiplication, i.e. σg ◦ (σf )−1 = σg◦(f −1 ) . Therefore, the set {fˆ | f ∈ Aut(X)} forms a subgroup of Aut(E1 (X)). Denote by ι the map that sends f ∈ Aut(X) to fˆ ∈ Aut(E1 (X)). If we identify the structures coming from S fin (X) in E1 (X) again with {Ai X | i ∈ ω}, the isomorphic copy of Aut(X) inside Aut(E1 (X)) consists of the subgroup of all f ∈ Aut(E1 (X)) such that f |X is in Aut(X) and f induces a permutation on {Ai | i ∈ ω}. It remains to show that ι : Aut(X) → Aut(E1 (X)) is a continuous embedding: Let ˆ f ∈ im(ι) be an automorphism of E1 (X). For an arbitrary finite subset a ⊆ E1 (X), let u := fˆ|a : a → E1 (X) be the restriction of fˆ to a and Ou := {g ∈ Aut(E1 (X)) | g|a = u} the basic open set defined by u containing fˆ. We have to show that the preimage of Ou under ι contains again an open subset. As a is finite, we can choose Ai1 , . . . , Ain  from above and C0 ⊆ X such that a is definable over C0 ∪ j=1,...,n Aij . Since the Aij correspond to finitely supported extensions of X, for each j = 1, . . . , n there exists a  Cj ⊆ X with Aij  | C X. Set by C := i≤n Ci and v := fˆ|C the restriction of fˆ to C. j We claim that Ov := {g ∈ Aut(X) | g|C = v} ⊆ Aut(X) is contained in the preimage

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of Ou under ι. Let g ∈ Ov be an arbitrary automorphism of X that extends v and gˆ its extension to E1 (X). As by assumption Aij  | C X and gˆ(C) = fˆ(C), Invariance implies gˆ(Aij )  | X and fˆ(Aij )  | X. fˆ(C)

(1)

fˆ(C)

Thus, Stationarity yields gˆ(Aij ) ≡X fˆ(Aij ). As both fˆ and gˆ are in the image of ι, the image of Aij under each of the two maps is again one of the Ak . On the other hand, every finitely supported extension of X has only been realized once within the Ak ’s, whence gˆ(Aij ) = fˆ(Aij ) for all j = 1, . . . , n. In particular, we get gˆ(a) = fˆ(a) and gˆ ∈ Ou . This proves that the embedding ι : Aut(X) → Aut(E1 (X)) is continuous. 2 With Lemmas 4.7 and 4.8 at hand, the main theorem now follows easily. Theorem 4.9 (Main theorem). Let M be a Fraïssé structure with stationary independence relation and Kω the class of all countable structures embeddable into M . Then for any X ∈ Kω , there is an embedding f : X → M such that every automorphism of f (X) extends to an automorphism of M in such a way that this extension yields a continuous embedding of Aut(X) into Aut(M ). In particular, the automorphism group Aut(M ) is universal for the class Aut(Kω ) := {Aut(X) | X ∈ Kω }, i.e. every group in Aut(Kω ) can be continuously embedded as a subgroup into Aut(M ). Proof. Assume X to be a Kω -structure. Lemma 4.8 yields that the automorphism group of X can be continuously embedded as a subgroup into the automorphism group of its Katětov limit E(X). As this limit is isomorphic to M by Lemma 4.7, we conclude that Aut(X) can also be continuously embedded as a subgroup into Aut(M ). 2 5. Generalized n-gons Several examples of Fraïssé structures with (local) stationary independence were already mentioned in the introduction, amongst them pure structures, relational Fraïssé limits with free amalgamation and the rational Urysohn space and sphere. In all these cases the construction given above yields that their automorphism groups are universal with respect to the class of automorphism groups of their countable substructures. Another SI-structure in a relational language, yet without free amalgamation, is the countable universal partial order P. Given finite partial orders A and B with a common substructure C, we define the amalgam A ⊗C B of A and B over C as the structure consisting of the disjoint union of A and B over C such that a ≤ b (resp. b ≤ a) if and only if there is some c ∈ C with a ≤ c ≤ b (resp. b ≤ c ≤ a). It is easy to check that the relation A  | C B defined by ABC ∼ = AC ⊗C BC provides a stationary independence relation on P.

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Further, non-relational examples have recently been provided by Baudisch [1], who shows that graded Lie algebras over finite fields and c-nilpotent groups of exponent p with an extra predicate for a central Lazard series are SI-structures, whence their automorphism group is universal. In the following, we will exhibit yet another homogeneous non-relational structure that admits a stationary independence relation: The countable universal generalized n-gon Γn , which arises as the Fraïssé limit of certain bipartite graphs [8]. A graph G = (VG , EG ) is bipartite if we can partition its vertex set VG = V1 ∪˙ V2 in such a way that every vertex in V1 is only connected to vertices in V2 and vice versa. We equip graphs with the graph metric dG , where dG (x, y) is the length of the shortest path from x to y in G. Furthermore, the diameter of G is the smallest number n ∈ N such that the distance between every two vertices x and y is at most n. If no such number exists, we say that G is of infinite diameter. Finally, the girth of G is the length of a shortest cycle in G. By a subgraph H ⊆ G we mean an induced subgraph. Definition 5.1. A generalized n-gon Γ is a bipartite graph of diameter n and girth 2n. Generalized n-gons were introduced by Jaques Tits, who developed the theory of buildings. For any n, a finite example is given by the ordinary n-gon, i.e. a cycle of length 2n. A projective plane provides an example of a generalized 3-gon. The class of generalized n-gons coincides with the class of spherical buildings of rank 2. We consider generalized n-gons Γ in the language Ln = P, fk | k = 0, . . . , n, where P is a predicate for the sort of the partition of the vertex set and the fk ’s are binary functions with fk (x, y) := xk if dΓ (x, y) = l ≥ k and there is a unique shortest path p = (x = x0 , . . . , xk , . . . , xl = y) from x to y. For l < n, such a unique path always exists, as otherwise there would be a non-trivial cycle of length 2l < 2n, contradicting the assumption on the girth of Γ. If there is no such unique path or dΓ (x, y) < k, we set fk (x, y) := x. Note that the edge relation is definable within this language as two vertices x and y are incident if and only if x = y and f1 (x, y) = y. Furthermore, if Δ ⊆ Γ is a generalized n-gon contained in Γ, then Δ is generated by a subgraph A ⊆ Δ as an Ln -structure, if and only if Δ is the smallest generalized n-gon in Γ containing A. Given any connected bipartite graph G without cycles of length less than 2n, we can build an Ln -structure inductively as follows: Set F0 (G) := G. Assume the graph Fi (G) has already been constructed. For any pair (x, y) in Fi (G) with distance n + 1 in Fi (G), we add a new path px,y = (x = x0 , x1 , . . . , xn−2 , xn−1 = y) from x to y of length n − 1, i.e. a path from x to y of length n − 1 such that all the xi for i = 1, . . . , n − 2 are new vertices. Clearly, the graph Fi (G) ∪ px,y still does not have cycles of length less than 2n, as every such cycle would have to contain the whole path px,y , but if we could complete px,y to a cycle of length less than 2n, there existed a path of length at most n between x and y in Fi (G), contradicting dFi (G) (x, y) = n + 1. Now we define

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Fi+1 (G) := Fi (G) ∪ {px,y | x, y ∈ Fi (G) and dFi (G) (x, y) = n + 1}  and call the graph F(G) := i∈ω Fi (G) the free n-completion of G. Observe that F(G) is a generalized n-gon. We now consider the class Cn of all free n-completions of finite connected bipartite graphs without cycles of length less than 2n and their Ln -substructures. Note that an arbitrary Ln -substructure of a generalized n-gon does not need to be a generalized n-gon itself: If we consider for example two points {x, y} of distance exactly n in an ordinary n-gon Δ, then there is no unique path of shortest length between them in Δ, whence the substructure generated by them consists merely of {x, y} without any edges. In particular, substructures of generalized n-gons need not be connected. Note though, that two vertices x and y in some generalized n-gon Δ are in different connected components in some Ln -substructure of Δ, only if their graph distance in Δ equals n. In particular, a non-connected Ln -substructure of Δ consists of vertices {xi | i ∈ I} such that the xi have mutual distance n in Δ. Tent shows in [8] that Cn is a Fraïssé class and hence it admits a countable homogeneous limit Γn . She also provides a characterization to recognize free n-completions of finite graphs via some corresponding weighted Euler characteristic. We will lift that characterization to infinite graphs and use it to define an independence relation on Γn . Definition 5.2. Consider the function χn on finite graphs H with vertex set VH and edge set EH defined by χn (H) := (n − 1)|VH | − (n − 2)|EH |. For arbitrary, possibly infinite graphs X ⊆ Y , we say that X is n-strong in Y (write X ≤n Y ) if and only if for all finite H ⊆ Y we have χn (H/H ∩ X) := χn (H) − χn (H ∩ X) ≥ 0. Lemma 5.3. Assume X ⊆ Y to be graphs. If Y arises from X by successively patching new paths of length n − 1, then the following hold: i) X ≤n Y and ii) if X ⊆ Y ⊆ Δ for some graph Δ with X ≤n Δ, then Y ≤n Δ. Proof. It suffices to consider the case where Y arises from X by adding one new path px,y = (x = x0 , x1 , . . . , xn−2 , xn−1 = y) of length n − 1 to two vertices x and y of X, i.e. Y := X ∪ px,y and xi ∈ / X for i = 1, . . . , n − 2. i) We have to show that χn (H/H ∩ X) = (n − 1)(|VH | − |VH∩X |) − (n − 2)(|EH | − |EH∩X |) ≥ 0

(2)

for any finite subgraph H ⊆ Y . Clearly, the inequality (2) holds, if |VH | − |VH∩X | ≥ |EH | − |EH∩X |, i.e. if there are more vertices in H outside of X than there are edges in H outside of X. Thus, as Y = X ∪ px,y , the inequality (2) could only fail, if H contained the entire path px,y . But then,

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χn (H/H ∩ X) = (n − 1)(|VH | − |VH∩X |) − (n − 2)(|EH | − |EH∩X |) = (n − 1)(|Vpx,y | − |{x, y}|) − (n − 2)|Epx,y | = (n − 1)(n − 2) − (n − 2)(n − 1) = 0. Hence, we get X ≤n Y , as desired. ii) Consider H ⊆ Δ finite. We have to show that χn (H/H ∩ Y ) ≥ 0. Let H  be the smallest subgraph of Δ that contains H and the path px,y . As Y = X ∪ px,y , all vertices in H  \ H are also in H  ∩ Y . This yields χn (H/H ∩ Y ) = (n − 1)(|VH | − |VH∩Y |) − (n − 2)(|EH | − |EH∩Y |) = (n − 1)(|VH  | − |VH  ∩Y |) − (n − 2)(|EH | − |EH∩Y |) ≥ (n − 1)(|VH  | − |VH  ∩Y |) − (n − 2)(|EH  | − |EH  ∩Y |) = χn (H  /H  ∩ Y ). One calculates as above that χn (H  ∩ Y ) = χn (H  ∩ X), because H  ∩ Y = (H  ∩ X) ∪ px,y . This yields χn (H/H ∩ Y ) ≥ χn (H  /H  ∩ Y ) = χn (H  /H  ∩ X) ≥ 0.

2

With Lemma 5.3 at hand, we can now prove the following characterization of free n-completions. The proof follows the proof of Proposition 2.5 from [8], which we included for completeness. Lemma 5.4. For any L-structure Δ ∈ Cn generated by a subset X ⊂ Δ, the following are equivalent: i) X is n-strong in Δ; ii) Δ is the free n-completion of X. Proof. ⇒: Assume X is n-strong in Δ. As above, we will denote by Fk (X) the k-th step of the free n-completion of X. We show that for all k we can embed Fk (X) as Xk over Xk−1 in Δ such that Xk ≤n Δ. As X generates Δ and F(X) is a generalized n-gon, this implies F(X) ∼ = Δ. For k = 0, the statement is given by the assumption, as F0 (X) = X. Now assume that Fk (X) ∼ = Xk ≤n Δ is isomorphic to Xk over X and consider x, y ∈ Xk arbitrary with dXk (x, y) = n + 1. As Δ is in Cn , the conditions on diameter and girth imply that there is a unique path px,y = (x, x1 , . . . , xn−2 , y) between x and y of length n − 1 in Δ given by xi = fi (x, y) for i ≤ n−2. We claim that xi ∈ / Xk for i = 1, . . . , n−2, i.e. that px,y is a new path. Note that necessarily one of the xi is outside of Xk , as otherwise dXk (x, y) = n − 1,

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contradicting the assumptions. Choose 1 ≤ j1 + 1 < j2 ≤ n − 1 such that xj1 , xj2 ∈ Xk and xl ∈ / Xk for all l with j1 < l < j2 . Let H := {xj1 , xj1 +1 , . . . , xj2 } be the path between xj1 and xj2 . Because Xk is n-strong in Δ, we know that χn (H/H ∩ Xk ) ≥ 0. An easy calculation shows that χn (H/H ∩ Xk ) ≥ 0 if and only if the path H has at least length n − 1, i.e. j1 = 0 and j2 = n − 1. This proves that px,y indeed is a new path and thus we can embed Fk+1 (X) as Xk+1 ⊆ Δ over Xk . By Lemma 5.3.ii) the structure Xk+1 is even n-strong in Δ, which concludes the induction. ⇐: If Δ is the free n-completion of X, it arises from X by successively patching new paths of length n − 1. Thus, Lemma 5.3.i) immediately implies that X ≤n Δ. 2 We can now define a stationary independence relation on Γn which corresponds to free amalgamation. For given graphs A, B and C with C ⊆ A, B, we denote the free amalgam of A and B over C by A ⊗C B. Definition 5.5 (Independence relation on Γn ). Let Γn be the countable universal homogeneous generalized n-gon as introduced above. For finitely generated substructures A, B and C ⊂ Γn , we define A and B to be independent over C (write A  | C B) if and only if the free amalgam AC ⊗C BC is n-strong in the substructure ABC ⊆ Γn generated by A, B and C in Γn . Lemma 5.4 yields that A  | C B if and only if the substructure generated by ABC is exactly the free n-completion of AC ⊗C BC. We will use both characterizations to prove the following main lemma. Lemma 5.6. The relation  | as stated in Definition 5.5 defines a stationary independence relation on Γn . Proof. Invariance and Symmetry are immediate and Existence follows since Γn is rich with respect to Cn -structures. Monotonicity: If A  | C BD, we know that AC ⊗C BCD ≤n ABCD.

(3)

We will first prove that this yields the equality ABC ∩ BCD = BC.

(4)

Recall that for any subgraph Γ ⊆ ABCD and points x and y in Γ with distance d (x, y) = n + 1, there is a unique path px,y = (x = x0 , x1 , . . . , xn−1 = y) ⊆ ABCD from x to y of length n − 1, by the conditions on diameter and girth in generalized n-gons. As before, we will denote by Fk := Fk (AC ⊗C BCD) the k-th step of the free n-completion of AC ⊗C BCD. As in the proof of Lemma 5.4 we know that we Γ

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can embed Fk as Xk over Xk−1 into ABCD with Xk ≤n ABCD. Inductively we define Γ0 := AC ⊗C BC and Γk+1 := Γk ∪ {px,y | x, y ∈ Γk , dXk (x, y) = n + 1}.   Note that Γk ⊆ ABC for each k and k∈ω Γk is a generalized n-gon, whence k∈ω Γk = ABC. Thus, in order to prove (4), it suffices to show that Γk ∩ BCD = BC for all k. We furthermore need to prove Γk ⊆ Xk , so that the calculation of the distance dXk (x, y) is well defined for arbitrary x and y in Γk . Both statements are clear for k = 0 by the inequality in (3). Now, assume that Γk ⊆ Xk and that Γk ∩ BCD = BC. Consider x and y in Γk with dXk (x, y) = n + 1. By the definition of a free n-completion, the unique path px,y of length n −1 between x and y will be contained in Xk+1 , whence Γk+1 ⊆ Xk+1 . As BCD ⊂ Xk and px,y ∩ Xk = {x, y}, we have px,y ∩ BCD = {x, y} ∩ BCD ⊆ Γk ∩ BCD = BC, and thus Γk+1 ∩ BCD = BC, as desired. Now, note that for arbitrary graphs X and Y , whenever X ≤n Y , then X ∩U ≤n Y ∩U for any U ⊆ Γk . Thus, for U = ABC, the equations (3) and (4) from above imply (4)

(3)

AC ⊗C BC = (AC ⊗C BCD) ∩ ABC ≤n ABCD ∩ ABC = ABC, so we got A  | C B, as desired. It remains to show that A  | BC D. Again by the equality in (4), we know that ABC ⊗BC BCD is contained in ABCD. Furthermore, the independence A  | CB implies that ABC ⊗BC BCD arises from AC ⊗C BCD by successively patching new paths of length n − 1. As AC ⊗C BCD ≤n ABCD, Lemma 5.3.ii) yields ABC ⊗BC BCD ≤n ABCD and thus A  | BC D as desired. Stationarity: Assume A1 , A2 , B and C are given such that A1 and A2 have the same type over C and are both independent from B over C. Then in particular, the graphs Ai C and BC form a free amalgam over C, whence there is partial isomorphism ϕ : A1 C ⊗C BC → A2 C ⊗C BC sending A1 to A2 while fixing BC. Clearly, the map ϕ extends to the free completions ϕ˜ : A1 BC → A2 BC. As Γn is homogeneous, the partial isomorphism ϕ˜ extends to an automorphism of Γn , which still fixes BC and maps A1 to A2 . Thus, the substructures A1 and A2 have the same type over BC. 2

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Corollary 5.7. The automorphism group of the countable homogeneous universal generalized n-gon Γn is universal for the class of all automorphism groups of generalized n-gons that are free over a finite subset. Acknowledgments I want to thank Amador Martin-Pizarro for guiding me into the topic and taking so much time to listen to first ideas and giving many helpful comments all along the way. I also want to thank Andreas Baudisch for fruitful discussions during my diploma thesis and Katrin Tent for suggesting the homogeneous generalized n-gons as a possible example and for comments about an earlier version of the paper. References [1] A. Baudisch, Free amalgamation and automorphism groups, J. Symbolic Logic (2016), to appear. [2] D. Bilge, Groupes d’automorphismes des structures homogènes, PhD thesis, Universite Claude Bernard–Lyon 1, 2012. [3] M. Doucha, Non-universality of automorphism groups of uncountable ultrahomogeneous structures, Topology Appl. 180 (2015) 209–217. [4] S. Gao, A.S. Kechris, On the classification of Polish metric spaces up to isometry, Mem. Amer. Math. Soc. 161 (2003) 78. [5] W. Hodges, A Shorter Model Theory, Cambridge University Press, New York, NY, USA, 1997. [6] E. Jaligot, On stabilizers of some moieties of the random tournament, Combinatorica 27 (1) (2007) 129–133. [7] M. Katětov, On universal metric spaces, in: General Topology and Its Relations to Modern Analysis and Algebra VI, 1988, pp. 323–330. [8] K. Tent, Free polygons, twin trees, and CAT(1)-spaces, Pure Appl. Math. Q. 7 (3) (2011) 1037–1052. [9] K. Tent, M. Ziegler, A Course in Model Theory, Cambridge University Press, 2012. [10] K. Tent, M. Ziegler, On the isometry group of the Urysohn space, J. Lond. Math. Soc. 87 (1) (Nov. 2012) 289–303. [11] V. Uspenskij, Compactifications of topological groups, in: Proc. of the 9th Prague Topological Symposium, 2001, pp. 331–346.